Summary

composite function: Let f and g be functions. The composition of g and f is the function whose values are given by g(f(x)) for all x in the domain of f such that f(x) is in the domain of g. g(f(x)) is read as ``g of f of x''.

If y is a function of u, say y=f(u), and if u is a function of x, say u=g(x), then y=f(u)=f(g(x)), and

or, using Leibniz's notation,

Both of these are forms of the chain rule. There is a chain of repercussions to a change in x: x changes, which leads to a change in g(x), which leads to a change in f(g(x)).

Note: the derivative of the composition is a product, both derivatives of f and g figuring in the product. The derivative of f has, as its argument, g(x). As the author says:

To find the derivative of f(g(x)),

1. find the derivative of f(x),
2. replace each x with g(x) [that is, write it as a composition], and
3. then multiply the result by the derivative of g(x).

Examples:

#7, p. 243

#9, p. 243

#29, p. 243

#53, p. 245